Here N is the tabulated difference in the logarithms corresponding to the difference 1 in the numbers at any part n of the table. Hence we can find N different numbers between n + 1 and n whose logarithms differ in the first 7 places of decimals. Hence, at this part of the table, we can find accurately to an Nth of unity any intermediate number whose logarithm is given correctly to 7 decimal places. Now N varies inversely as n. Hence, as we proceed in the table, we can interpolate, between numbers differing by the constant amount unity, a continually smaller number of values corresponding to given logarithms calculated to a fixed number 7 of decimal places. 566. To find the difference of the sines of two angles approximately in terms of the difference of the angles. Let 8<1 be the circular measure of an angle. Then sin (a +8) — sin a = cos a sin 8 - sin a (1 − cos 8) = cos a (8 – 183 + ...) — sin a (182–21484 + ...). In this series after 8 cos a the terms, taken in pairs, are continually decreasing and alternate in sign. Hence the series differs from 8 cos a by a quantity less than 182 sin a + 183 cos a. Let & be the circular measure of an angle less than l'. Then Hence, if 8 is not greater than the circular measure of 1', the equation sin (a + 8) - sin a = 8 cos a 26 holds as far as 7 decimal places. J. T. 567. To interpolate the sine of any given angle between a and a+l', we may use the above formula which will give the required sine correctly to 7 decimal places. 568. But conversely, to interpolate an angle corresponding to a given sine, in the sine-difference formula put for 8 the circular measure of 1'. Then sin (a + 1') - sin a = π 10800 COS α= - N. 10-7 say. Here N is the tabulated difference in the sines corresponding to the difference l' in the angles at any part a of the table. Hence we can find N different angles between a + l' and a whose sines differ in the first 7 decimal places. Now as the angle a increases up to 90°, cos a and therefore N decreases. Hence we can find intermediate angles from their sines with continually decreasing accuracy as we approach 90°. For example, sin 78° 5′ and sin 78° 6' differ by 600.10-7. Hence up to 78° 6' we can, from the sines of angles given correctly to 7 places, interpolate angles correctly to within a 600th of a minute; i.e. a 10th of a second. Again sin 84° 3′ and sin 84° 4' differ by 300. 10-7. Hence up to this point we can similarly interpolate correctly to within a 5th of a second. And so on. 569. To find the difference of the tangents of two angles approximately in terms of the difference of the angles. - Thus tan (a +8) – tan a differs from 8 sec2 a by a quantity less than This quantity increases as a and as 8 increase. Now it will be found by examining the tables that approximately tan a sec2 a = 1, when a is 34°; = 10, when a is 64°; = 100, when a is 78°; = = 1000, when a is 84°: and so increases more and more rapidly up to 90°, when it is infinite. Hence the tangent-difference formula is less and less accurate as the angle approaches 90°. 570. Hence, to interpolate a tangent corresponding to a given angle between a and a + 1', the equation tan (a +8) - tan a = 8 sec2 a holds for 7 places, up to 34°; for 6 places, up to 64°; for 5 places, up to 78°; for 4 places, up to 84°; and so on. 571. Conversely, to interpolate an angle corresponding to a given tangent, in the tangent-difference formula put for the circular measure of 1'. Thus tan (a + 1') - tan a = П 10800 sec2 a = N. 10-7 say. Here N, which varies as sec2 a, increases as a increases. Its smallest value, viz. the difference between tan 0 and tan l', is about 3000. Hence even here we could find an angle correctly to within nearly of a second corresponding to a tangent given correctly to 7 decimal places. 1 50 Now the difference in the tangents for l' is at 34° about 4000 x 10-7; at 64°, 1500 x 10-6; at 78°, 600 x 10-5; at 84°, 270 × 10-4; and so on. 1 Hence we can find angles, when the tangents are given correctly to 7 decimal places from 0 to 34°, with an accuracy increasing from to of a second: when the tangents are given to 6 places from 34° to 64°, with an accuracy increasing fromto of a second: when the tangents are given to 5 places from 64° to 78°, with an accuracy increasing from to of a second: when the tangents are given to 4 places from 78° to 84°, with an accuracy increasing from 1 to of a second and so on. 572. 25 To find the difference of the secants of two angles approximately in terms of the difference of the angles. Here we have sec (a + 8) - sec a = sec a sec & 1- tan a tan d sec a = sec a sec 8 (1 + tan a tan 8 + tan2 a tan2 8+ ...) — sec a. Now sec 81 and tan 8 > 8. secants sec a tan a. But cos 8>1-182 and sin 8<8. Hence the difference of the Hence the difference of the secants sec a sec a. 1-8 tan a *d Hence sec (a + 8) - sec a differs from 8 sec a tan a by a quantity This quantity increases as a and as 8 increase. Now it will be found by examining the tables that approximately sec a (sec2 a — 1)=1 when a is 33°; = 10 when a is 63°; = 100 when a is 77°; = 1000 when a is 84°; and so increases more and more rapidly up to 90° where it is infinite. Hence the secant-difference formula is less and less accurate as we approach 90°. 573. Hence, to interpolate a secant corresponding to a given angle between a and a + 1', the equation sec (a + 8) - sec a = 8 sec a tan a holds for 7 places up to 33°; for 6 places up to 63°; for 5 places up to 77°; for 4 places up to 84°; and so on. 574. Conversely, to interpolate an angle corresponding to a given secant, in the secant-difference formula put for 8 the circular measure of 1'. Thus π sec (a + 1') - sec a = sec a tan a = 10800 Here N increases as a increases. Hence, as we approach zero, we can interpolate an angle with less and less accuracy corresponding to a given secant. For instance sec 11° 28′ and sec 11° 29' differ by 600 × 10−7. Hence here we can find angles to within of a second corresponding to secants given correctly to 7 places. Again sec 3° 13' and sec 3° 14' differ by 160 × 10-7. Hence here we can interpolate an angle correctly to within only of a second. And so on. On the other hand as we approach 90° we find the following results. The difference in the secants for 1' is at first about 2 × 10-7, and at 33° about 2000 × 10-7; at 63°, 1200 × 10-6; at 77°, 600 x 10-5; at 84°, 250 × 10-4; and so on. Hence we can find angles, when the secants are given correctly to 7 decimal places from 1' to 33°, with an accuracy increasing from 30" to of a second; when the secants are given correctly to 6 decimal places from 33° to 63°, with an accuracy increasing from to of a second; when the secants are given correctly to 5 decimal places from 63° to 77°, with an accuracy increasing fromto of a second; when the secants are given correctly to 4 decimal places from 77° to 84°, with an accuracy increasing from 1 to of a second; and so on. 575. The secondary ratios cosine, cotangent, and cosecant of an angle being simply the sine, tangent, and secant respectively |